154 JOURNAL OF THE WASHINGTON ACADEMY OF SCIENCES VOL. 13, No. 8 
or; putting 
Sf i — a 98 ob (15) 
and fusing constants in obvious notation 
a= -y+ Bxy+Cy?+ Ex’y+Fxy?+Gy?+...=M (16) 
oy yee A’x? + B’xy + D’x? + E’x*y + F’xy?+...=N (17) 
Consider now the function 
3 
e=x-+ty?+2(A'+B) = —2(B'+C) x +2[(A’+B)B+E+D'] = 
+2[(B’+C)B'-G-¥) ¥ (18) 
With the function thus defined let us form the expression 
dg Oy dx ree dy M22 rey) 
Bp sees DEO ye. we 
ad). yexdT ss oy dT Ox i oy (19) 
It is thus found that 
we = 2(BC — AB! + E’ + F) xy? +8 (20) 
-(— i =) xy? +S =Rxy?+S8 (21) 
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where S contains only terms of 5th and higher degree. So long as 
x, y, do not exceed a certain value, the term of fourth degree is the 
one that determines the sign of “= In that case evidently, 
<= 0 according as ig =o, 1.€, , ase (22) 
Now consider the family of curves defined by 
¢ = constant = K (23) 
It is clear from (18) that in the immediate neighborhood of the origin 
these are concentric circles of radius yK. More generally, near the 
origin, they are closed curves enclosing the origin, and such that the 
curve ¢ = K, completely encloses (without contact) the curve ¢ = K,, 
if K,>K, 
